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We investigate conditions under which, for two sequences $(u_r)$ and $(v_r)$weakly converging to $u$ and $v$ in $L^p(R^d;R^N)$ and $L^{q}(R^d;R^N)$,respectively, $1/p+1/q \leq 1$, a quadratic form$q(x;u_r,v_r)=\sum\limits_{j,m=1}^N q_{j m}(x)u_{j r} v_{m r}$ converges toward$q(x;u,v)$ in the sense of distributions. The conditions involve fractionalderivatives and variable coefficients, and they represent a generalization ofthe known compensated compactness theory. The proofs are accomplished using arecently introduced $H$-distribution concept. We apply the developed techniquesto a nonlinear (degenerate) parabolic equation.

On a generalization of compensated compactness in the $L^p-L^q$ setting